Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•3x213) • x) -  5  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  6x²¹⁴-5 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  6  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Equation at the end of step  2  :

  6x214 - 5  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    6x²¹⁴-5 = 0 

 Add  5  to both sides of the equation : 
                      6x²¹⁴ = 5
Divide both sides of the equation by 6:
                     x²¹⁴ = 5/6 = 0.833
                     x  =  214th root of (5/6) 

 The equation has two real solutions  
 These solutions are  x = 214th root of ( 0.833) = ± 0.99915  
 

Two solutions were found :